(c) 2007 IUPUI SPEA K300 (4392) Outline: Numerical Methods Measures of Central Tendency Representative value Mean Median, mode, midrange Measures of Dispersion.

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(c) 2007 IUPUI SPEA K300 (4392) Outline: Numerical Methods Measures of Central Tendency Representative value Mean Median, mode, midrange Measures of Dispersion (Variability) How are data points are deviated from the mean? Range variance, standard deviation

(c) 2007 IUPUI SPEA K300 (4392) Central Tendency: Mean Arithmetic average Sum divided by N (# of observations) Key statistic in data analysis

(c) 2007 IUPUI SPEA K300 (4392) Central Tendency: Mean ClassMidpointFrequencyFrequency × Midpoint Sum Question 12 on page 117 The mean is = / 108

(c) 2007 IUPUI SPEA K300 (4392) Central Tendency: Median Midpoint of data arranged in order When even number of observations, mean of the two data points in the middle Useful when data are skewed to the right or left substantially.

(c) 2007 IUPUI SPEA K300 (4392) Central Tendency: Mode Value that occurs most often Peak in the histogram Bimodal with two peaks (modes) Figure 3-1 on page 115

(c) 2007 IUPUI SPEA K300 (4392) Central Tendency: Midrange Mean of minimum and maximum values (minimum + maximum)/2

(c) 2007 IUPUI SPEA K300 (4392) Central Tendency: Others Weighted mean when individual data points should be weighted differently (other than 1) Trimmed mean in the presence of outliers (extremely large or small data points)

(c) 2007 IUPUI SPEA K300 (4392) Quantiles Quantiles are points taken at equal intervals from CDF (cumulative density function) 100 quantiles: percentiles 10 quantiles: dociles 5 quantiles: quintiles 4 quantiles: quartiles 2 quantiles: ?

(c) 2007 IUPUI SPEA K300 (4392) Percentiles Percentiles divide data into 100 groups with an equal interval 100 quantiles Nth percentile is located at nth from the smallest in data w/ 100 observations Median = 50 th percentile Table 3-3 on page 141 Figure 3-5 on page 142

(c) 2007 IUPUI SPEA K300 (4392) Quartiles 4-quantiles 1 st quartile (25 th percentile) 2st (50 th percentile or median) 3st (75 th percentile) IQR (interquartile range) = 3Q-1Q Box plot include 1Q, 2Q, 3Q, minimum, maximum

(c) 2007 IUPUI SPEA K300 (4392) Quartiles in a Box Plots

(c) 2007 IUPUI SPEA K300 (4392) Quartiles in Histograms

(c) 2007 IUPUI SPEA K300 (4392) Example 1: Example 3-38, p159

(c) 2007 IUPUI SPEA K300 (4392) Example 2: SAS Output SAS includes mean in the box plot Histogram # Boxplot 2.75+* 1 |.** 4 |.******** 23 |.**************** 46 |.*********************** *************************** 80 | |.*************************************** 116 *--+--*.********************** ******************* 56 |.********* 27 |.***** 13 | * 2 | * may represent up to 3 counts

(c) 2007 IUPUI SPEA K300 (4392) Example 3: Box Plots | 9 + | | | | | 8 + | *-----* | | | | | | | + | | | | | | | | | 6 + | | | | | | | | 5 + | | | | + | | | | | | | | 4 + *-----* | | | | | | | | | | | + | | | | | | *-----* | | | | 2 + | | | | 1 + | | Site